Is an infinite multitude self-contradictory?

Philosophy Forums Philosophy
1

Dear cyberphilosophers (not to say: “truth-seekers”, because you don’t even believe in truth),

          I wonder now whether an infinite multitude is possible. Suppose there is such a multitude. I separate five elements from the rest. So I have two sets: the set A, with five elements, and the set B, with all other elements. My question is: is there an infinite multitude in the set B, even after the removal of the five elements?

There are only two possibilities: the set B contains an infinite multitude or a finite multitude. If the set B an infinite multitude, it means that Infinite = Infinite - 5, therefore that a quantity is inferior to itself… It can’t be that.

If there is a finite multitude, it means that: infinite - 5 = finite. But we can change the place of the elements of the formula and thus we get: finite + 5 = infinite. In other words, we are supposed to believe that the infinite consists in the addition of two finite quantities. It is impossible. So it can’t be that either.

It can be objected that what is wrong with the above reasoning is the idea of “set”. The infinite multitude can’t be seen as a set. If we separate five elements from their surroundings, what is outside these elements does not make up a “set”. There is no such a thing as a set B.

If the idea of the set does not apply, neither do the idas of addition and of subtraction, because these operations are defined with reference to a set.

However, it seems that a multitude which does not make up a set is meaningless. It does not make any sense.

Let’s have a look at the Webster dictionary:

Multitude: “a large number; crowd”

So let’s have a look at “number” and at “crowd”.

Number: “the sum of a collection of persons or things; total”

Crowd: “a large number of people or things gathered closely together”

It seems that we come across synonyms of a set: “number”, “collection”, “total”.

The burden of proof is thus upon those who claim that an infinite multitude makes sense.

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Well, I would be very grateful to you if you took the time to post your comments on this problem, which has deep consequences on the nature of reality… :slight_smile:

3

infinite multitude?

ok… the distance between point 0 and 1 on a number line can be infinitely halved and the sum of the series of infinite halves would then be 1 (or as close to 1 as infinitely possible)… same goes for the distance between 1 and 2, 2 and 3, 3 and 4… ect. …

infinite multitudes are demanded by division…

-Imp

4

Thanks for your reply, Imp.

I think that the relevant word in your comment is “can”. The line between 0 and 1 is just a potential infinite. And moreover, a potential infinite in the abstract realm of numbers… In the realm of physical things, I don’t believe that there can be an infinite multitude of parts between two points. The important thing to know is that even if a thing is extended, it does not entail necessarily that it can be divided. There must be a limit to the possibility of division of a whole in the physical realm.

Thus Lucretius proved the existence of atoms with a paradox on the infinite division…

5

but the boys at los alamos split the atom…

-Imp

6

What they call an atom and what Lucretius called an atom are not the same thing :wink:

But Dalton, the chemist who borrowed the term “atom”, thought indeed that the atom was impossible to divide…

7

Your question has been answered for over a century. The book to read is anything on introductory set theory, in particular Cantor set theory. The problems with your argument are

  1. addition and subtraction do not apply in the ordinary way to infinite sets
  2. they remain sets despite this fact, because being a set does not depend on obeying the ordinary rules of addition and subtraction on finite sets

In the future, please direct all mathematical and physical questions directly to my office, and fill out form F-1…

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Thus in what sense do addition and subtraction apply to infinite sets?

And what is the definition of a set?

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Rather than just tell you to search Wiki, I think I can remember hw it was explained to me:

Infinite sets, even when subtracted or added, are the same size (cardinality it’s called)

For your example, lets put the two sets into a 1-to-one correpondance (but I’m using infinity = infinity -1 for ease)

…-1, 0, 1… with inifinitely many members either side

the other is:

…0, 1, 2…

But when you think about it, they both contain -1 as well as both containing 2 too, so they can be mapped exactly one-to-one, therefore they’re the same size.

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I am speaking of an infinite number in the physical world, not in the mathematical realm, where negative numbers exist…

Besides, your sequence is not a true subtraction: one sequence is merely delayed with respect to the other one.

It’s like… if we are at the library, and if we go to the park, but whereas I go there at noon and you go there at three o’clock, so we will have covered the same number of kilometers, though the description of our route will look like this:

         Noon        3 o'clock        6 o'clock        9 o'clock

Me 0 km 10 km 20 km (arrival) 20 km

You 0 km 0 km 10 km 20 km (arrival)

There is no real subtraction, but merely a delay. But in the physical world, there is nonetheless the possibility of real subtractions.

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Whatever answer you give to the paradox I raise, you have to explain how this answer applies to the physical world.

In the physical world, there are only integers. There are no negative numbers. Zero means nothingness. Etc.

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Most of the rules you learned in elementary school for manipulating equations and inequalities do not apply to infinite sets. Modern mathematicians from the late 19th century onward went to the trouble of proving those laws for finite sets from very basic axioms – axioms similar to the Zermelo-Frankel with Choice (ZFC) axioms mathematicians use today. The proofs work because of certain properties of finite sets that infinite sets do not have. Trying to apply ordinary addition, subtraction, and inequality signs to infinite sets leads to nonsense quickly. For example, consider the following reasoning:

The set {1, 3, 5…} and the set {2, 4, 6…} are infinite, and so is the set {1, 2, 3, 4…}. Note that the first set plus the second set equals the third set. Therefore we may write inf + inf = inf. Subtracting inf from both sides we get inf = 0.

The reason we erred here is because the ordinary laws of arithmetic no longer apply. In fact, ordinary arithmetic is mostly useless when dealing with infinite sets.

13

The sets you give as examples are not infinite sets. It creates only the illusion of an infinite set. It is the confusion in our mind that makes us think that there can be infinite sets. In fact, there are only as many elements in the sets as you can think. The set [1, 2, 3…] contains in fact three elements, but you add the points, which merely mean that it is up to us to enlarge the set. But however you enlarge the set, there is never an actual infinite.

It’s like a propeller of a plane which revolves quickly. It looks like a circle, but it is not. The illusion of the circle arises out of our confusion…

Perhaps you could suggest to me a book on philosophy of maths which requires little background knowledge…?

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Space from Zeno to Einstein -Nick Huggett

Infinity and the Mind- Rudy Rucker

-Imp

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And how are these books related to the current issue? How can they help us settle the problem? Don’t just say: “they talk about infinite”. Tell me what they say that can be relevant.

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the huggett book walks you through the philosophers who discussed these problems in several dimensions and the rucker book explores the concept of infinity on many levels, particularly infinite sets…

mitpress.mit.edu/catalog/item/de … 2&tid=4115

pupress.princeton.edu/titles/5656.html

-Imp

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Impenitent

 But it's still 3 inches (or whatever)- a finite amount. I think we all agree that infinities can exist in abstract mathematics, but the potential of something to be devided isn't an example of an infinity existing [i]in reality[/i]. It would be like saying that if I never died, I would live an infinite amount of time- Technically true, but at any point in reality, my span would be finite.  Unless this 'having' is something that is actually happening, I don't see how this shows that infinities exist in reality.
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No, I am not so sure. Every time someone gives an example of an infinite set, he puts some numbers and then three points forward or backward, but it is not a proof… It is only assumed that the points point to an infinite set. It is as if I said that since I have the concept an infinite being, this being must exist… The concept of the infinite (if there is such a concept to begin with) may be void…

It is an old assumption that there exist numbers independently of the mind, that these numbers are objective and eternal, but what I think is that, perhaps, a number does not exist before someone thinks about it…

19

ive always thought the simple solution to this problem and others like it is that infinity is not a number but the characteristic of never ending. a million is a number, but infinity just means that you dont stop at a million or anything.

the only times i remember this coming up are when physics cant explain things like the big bang and black holes. they dont know when those things are going to stop getting smaller so they just say infinity. or rather, their amazing math has an asymptote in the graph of the equation.

its just not a number, its a characteristic of a thing that may sometimes be described by numbers, and an indication that if our current theory is taken to a certain point, it is no longer an accurate description.

20

Thus it seems that infinity is a subjective feature for you…? Infinity qualifies how we deal with numbers, and not one of these numbers…

So I am right: an infinite set of physical things is impossible.